1. Divisibility and Modular Arithmetic
Section 4.1

2. Division
Definition: If a and b are integers with a ≠ 0, then a divides b if there exists an integer c such that b = ac.
The notation a | b denotes that a divides b.
If a | b, then b/a is an integer.
If a does not divide b, we write a ∤ b.
Example: Determine whether 3 | 7 and whether | 12.

3. Properties of Divisibility
Theorem: Let a, b, and c be integers, where a ≠0.
If a | b and a | c, then a | (b + c);
If a | b, then a | bc for all integers c;
If a | b and b | c, then a | c.
Proof: (i) Suppose a | b and a | c, then it follows that there are integers s and t with b = as and c = at. Hence,
b + c = as + at = a(s + t). Hence, a | (b + c)
Corollary: If a, b, and c be integers, where a ≠0, such that a | b and a | c, then a | mb + nc whenever m and n are integers.

4. Definitions of Functions div and mod
Division Algorithm
When an integer is divided by a positive integer, there is a quotient and a remainder. This is traditionally called the “Division Algorithm,” but is really a theorem.
Division Algorithm: If a is an integer and d a positive integer, then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r
d is called the divisor.
a is called the dividend.
q is called the quotient.
r is called the remainder.
Examples:
What are the quotient and remainder when 101 is divided by 11?
Solution: The quotient when 101 is divided by 11 is 9 = 101 div 11, and the remainder is 2 = 101 mod 11.
What are the quotient and remainder when −11 is divided by 3?
Solution: The quotient when −11 is divided by 3 is −4 = −11 div 3, and the remainder is 1 = −11 mod 3.
Definitions of Functions div and mod
q = a div d
r = a mod d

5. Congruence Relation
Definition: If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a–b.
The notation a ≡ b (mod m) says that a is congruent to b modulo m.
Two integers are congruent mod m if and only if they have the same remainder when divided by m.
Example: Determine whether 17 is congruent to 5 modulo 6 and whether 24 and 14 are congruent modulo 6.
Solution:
17 ≡ 5 (mod 6) because 6 divides 17 − 5 = 12.
24 ≢ 14 (mod 6) since 6 divides 24 − 14 = 10 is not divisible by 6.

6. More on Congruences
Theorem: Let m be a positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km.
Proof:
If a ≡ b (mod m), then (by the definition of congruence) m | a – b. Hence, there is an integer k such that a – b = km and equivalently a = b + km.
Conversely, if there is an integer k such that a = b + km, then km = a – b. Hence, m | a – b and a ≡ b (mod m).

7. Congruences of Sums and Products
Theorem: Let m be a positive integer. If a ≡ b (mod m) and c ≡ d (mod m), then
a + c ≡ b + d (mod m) and ac ≡ bd (mod m)
Proof: (good exercise)
Because a ≡ b (mod m) and c ≡ d (mod m), by previous theorem there are integers s and t with b = a + sm and d = c + tm.
Therefore,
b + d = (a + sm) + (c + tm) = (a + c) + m(s + t) and
b d = (a + sm) (c + tm) = ac + m(at + cs + stm).
Hence, a + c ≡ b + d (mod m) and ac ≡ bd (mod m).
Example: Because 7 ≡ 2 (mod 5) and 11 ≡ 1 (mod 5) , it follows from that
18 = ≡ = 3 (mod 5)
77 = (7)11 ≡ (2)1 = 2 (mod 5)

8. Arithmetic Modulo m
Definitions: Let Zm be the set of nonnegative integers less than m: {0,1, …., m−1}
The operation +m is defined as a +m b = (a + b) mod m. This is addition modulo m.
The operation ∙m is defined as a ∙m b = (ab) mod m. This is multiplication modulo m.
Using these operations is said to be doing arithmetic modulo m.
Example: Find and 7 ∙11 9.
Solution: Using the definitions above:
= (7 + 9) mod 11 = 16 mod 11 = 5
7 ∙11 9 = (7 ∙ 9) mod 11 = 63 mod 11 = 8

9. Arithmetic Modulo m
The operations +m and ∙m satisfy many of the same properties as ordinary addition and multiplication.
Closure: If a and b belong to Zm , then a +m b and a ∙m b belong to Zm .
Associativity: If a, b, and c belong to Zm , then (a +m b) +m c = a +m (b +m c) and (a ∙m b) ∙m c = a ∙m (b ∙m c).
Commutativity: If a and b belong to Zm , then a +m b = b +m a and a ∙m b = b ∙m a.
Identity elements: The elements 0 and 1 are identity elements for addition and multiplication modulo m, respectively.
If a belongs to Zm , then a +m 0 = a and a ∙m 1 = a.
continued →

10. Arithmetic Modulo m
Additive inverses: If a≠ 0 belongs to Zm , then m− a is the additive inverse of a modulo m and 0 is its own additive inverse.
a +m (m− a ) = 0 and 0 +m 0 = 0
Distributivity: If a, b, and c belong to Zm , then
a ∙m (b +m c) = (a ∙m b) +m (a ∙m c) and (a +m b) ∙m c = (a ∙m c) +m (b ∙m c).
Multiplicatative inverses have not been included since they do not always exist. For example, there is no multiplicative inverse of 2 modulo 6.
(extra) Using the terminology of abstract algebra, Zm with +m is a commutative group and Zm with +m and ∙m is a commutative ring.